Place the chicken chunks in a small bowl and sprinkle with 1 teaspoon of the salt and 0.25 teaspoon of the black pepper. Add 1 tablespoon of the flour and toss until every piece is lightly and evenly coated.
Heat 1 tablespoon of the olive oil in your Dutch oven over medium-high heat until it shimmers. Add the chicken in a single layer, working in two batches if needed, and sear undisturbed for about 3 minutes per side until golden. Remove to a plate and set aside.
Reduce the heat to medium and add the remaining 1 tablespoon olive oil to the same pot. Add the diced onion, carrots, and celery, then cook for about 5 minutes, stirring occasionally, until they begin to soften. Scrape up any browned bits, then stir in the minced garlic and cook for 1 more minute until fragrant.
Add the 3 tablespoons of butter to the vegetables and let it melt completely. Sprinkle in the remaining 2 tablespoons of flour and stir constantly for about 2 minutes until the flour loses its raw smell and coats the vegetables in a pale paste.
Pour in the chicken broth slowly, stirring the whole time to dissolve the roux and prevent lumps. Add the cubed potatoes, dried thyme, crushed rosemary, and bay leaves. Return the browned chicken and any juices to the pot, then stir in the remaining 0.5 teaspoon salt and 0.25 teaspoon black pepper.
Bring the stew to a gentle boil, then reduce the heat to low. Cover the pot and simmer for 20 minutes, stirring once or twice, until the potatoes are fork-tender and the chicken is cooked through to 165°F (74°C).
Uncover the pot and let the stew simmer for 5 to 10 more minutes to reduce and thicken the broth. Remove and discard the bay leaves. Turn off the heat and stir in the heavy cream until the broth turns silky. Taste and adjust salt and pepper, then stir in the fresh parsley just before serving.